可以发现这些概念的约束是递进的
A semigroup is a magma where the binary operator must be associative.
- In other words, a semigroup is a set with a binary operator that is closed and associative.
集合+二元操作符,满足结合律+封闭性
A monoid is a semigroup with an identity element.
在semigroup 基础上加上了 identity element. 特性
monoid plus below opration
- inverse
Cyclic Group
存在几何中的一个元素g,对g不断进行群运算得到的元素,可以生成整个群;
- elliptic curve over a finite field(eg. integer mod p) is Cyclic Group
满足交换律的group
A Ring is a set with two binary operators such that
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under the first binary operator, the set is a abelian group
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under the second binary operator, the set is a monoid
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the second binary operator distributes over the first
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example
Prove:A ring with only {0} under addition and multiplication is a trivial ring
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多项式是一个环:在加法下满足abelian group,乘法下满足monoid(没有逆运算)
A field is a set with two binary operators such that
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under the first binary operator, the set is an abelian group
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under the second binary operator, excluding the zero element, the set is an abelian group
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The set of all integers under addition and multiplication is not a field
- 因为在乘法下面,整数的逆可能不再是整数(eg.5)
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The set of all rational numbers is a field
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Integers modulo a prime number is a field under addition and multiplication
- in modular arithmetic, every element in a finite field has a multiplicative inverse;
- 关键点来了!!! 后续可以经常看到很多mod操作,原因就是mod操作之后满足filed条件
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有限域(finite field)